Continuous dependence for NLS in fractional order spaces

TL;DR

This paper proves that solutions to the nonlinear Schrödinger equation depend continuously on initial data in fractional Sobolev spaces, with Lipschitz dependence when the nonlinearity exponent is at least one, extending known results.

Contribution

It establishes continuous dependence of solutions in fractional Sobolev spaces for the NLS, including Lipschitz dependence for certain nonlinearities, filling gaps in previous regularity cases.

Findings

Continuous dependence in $H^s$ for $0<s<1$

Lipschitz dependence when $ ext{exponent} \\ge 1$

Extension of results for $s=0,1,2$ cases

Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger equation $iu_t+ \Delta u+ \lambda |u|^\alpha u=0$ in $\R^N $, in the $H^s$-subcritical and critical cases $0<\alpha \le 4/(N-2s)$, where $0<s<N/2$. Local existence of solutions in $H^s$ is well known. However, even though the solution is constructed by a fixed-point technique, continuous dependence in $H^s$ does not follow from the contraction mapping argument. In this paper, assuming furthermore $s<1$, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous $H^s \to H^s$. If, in addition, $\alpha \ge 1$ then the flow is Lipschitz. This completes previously known results concerning the cases $s=0,1,2$.